2003/12/10 by Volodymyr Batchenko, Fritz Gesztesy, Batchenko, Volodymyr +1 · 1 citation
Mathematics · Physics and Astronomy · #34L05 #34L40 #35Q51 #35Q53 #58F07 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:34L05 #msc:34L40 #msc:35Q51 #msc:35Q53 #msc:58F07 #nlin.SI
paper · pdf · doi:10.48550/arxiv.math/0312200
43 pages
arxiv created 2003/12/10 · openalex publication_date 2003/12/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We characterize the spectrum of one-dimensional Schrödinger operators H=-d2/dx2+V with quasi-periodic complex-valued algebro-geometric potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The corresponding problem appears to have been open since the mid-seventies. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green's function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well.